Integral geometry on affine buildings (Q2704083)

This paper contributes to the theory of Radon transforms on affine buildings. In fact, it is a kind of survey paper of earlier published work of the author [J. Aust. Math. Soc., Ser. A 66, 66-89 (1999; Zbl 0938.51005) and Math. Proc. Camb. Philos. Soc. 128, 425-439 (2000; Zbl 0966.51006)]. She uses this occasion to derive a better and more explicit inverse formula in case of Radon transforms on buildings of type \(\widetilde C_2\). It is based on the fact that one can count the number of apartments contained in the geometry opposite a given chamber of a generalized quadrangle (this is no longer true for hexagons, and so a similar result for buildings of type \(\widetilde G_2\) seems not within reach, except if one assumes that the relevant residues are of known type). She also discusses the dual Radon transform, and obtains explicit expressions for inverting this in case of affine buildings of type \(\widetilde A_2\).NEWLINENEWLINENEWLINEFinally, she introduces horocyclic Radon transforms. Roughly speaking, a horocycle is the inverse image of a vertex for the projection of the vertex set of an affine building onto an apartment relative to some chamber at infinity contained in that apartment. Then the horocyclic transform of a real or complex function on the vertices of an affine building maps a horocycle to the sum of all function values of vertices contained in the horocycle. For trees (affine buildings of rank 2), these transforms are already studied in the literature, and the author intends to study these in the general case (and she announces some results to be proved in a forthcoming paper).NEWLINENEWLINENEWLINEThe paper contains an elementary introduction to buildings, which should appeal to readers not familiar with buildings; but the paper is harder to read for people familiar with buildings, but not with Radon transforms. Moreover, the paper is not proofread very well, so annoying typos keep popping up (as, for instance, a reference to Proposition 1.2 which does not exist).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00041].